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\title{Transfer of unitary representations}
\author{Ma Jia Jun}

\begin{document}

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\def\gg{{\mathfrak{g}}}
\def\kk{{\mathfrak{k}}}
\def\pp{{\mathfrak{p}}}
\def\qq{{\mathfrak{q}}}
\def\oo{{\mathfrak{o}}}
\def\mm{{\mathfrak{m}}}
\def\bC{{\mathbb{C}}}
\def\bR{{\mathbb{R}}}
\def\hK{{\widehat{K}}}
\def\cC{{\mathcal{C}}}
\def\Hom{\mathrm{Hom}}
\def\inn#1#2{\left\langle{#1},{#2}\right\rangle}
\def\innc#1#2{\left({#1},{#2}\right)}

\def\tt#1{{\color{yellow}#1}}
\begin{frame}[t]
\begin{center} 
\large Transfer of unitary representations
\end{center}
\begin{center}
 Ma Jia Jun
\end{center}
\end{frame}

\begin{frame}[t]
\tt{Notation:} $G_\bC$ simply connected, semi-simple Lie group over $\bC$.
$G,G'$ real reductive group with cartan involution 
$\theta,\theta'$ and maximal compact subgroup
$K,K'$, $M=K\cap K'$,
$\sigma,\sigma'$ complex conjugation respect to $\gg_0, \gg'_0$.
\end{frame}

\begin{frame}[t] \tt{$(\gg,K)$-module:} $\bC$-vector spece with $\gg$ action and finite $K$ action
satisfing certain compatibility condition. Admissible:
$\dim \Hom_K(\tau,V) < \infty, \forall \tau\in \hK$. 
Study irr. $G$-rep can be reduce to study adm. 
$(\gg,K)$-module. $(\gg,K)$-module form a categray $\cC(\gg,K)$.
\end{frame}

\begin{frame}[t]
\tt{Zuckerman functors:} $\Gamma:\cC(\gg,M)\to \cC(\gg,K)$ 
\[
\Gamma(V) = \Set{K\text{ finite vectors in V}}
\]
\[
(\Gamma_{M,K})^i=R^i\Gamma: \cC(\gg,M)\to \cC(\gg,K)
\]
\end{frame}

\begin{frame}[t]
(A1) $\kk$ hase a real form $\kk_1$ and $(K_1,M)$ is a symmetric pair.\\
(A2) $V = \bigoplus_j L_j$, $L_j$ irreducible unitarizable $(\kk,M)$-module.
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\begin{frame}[t]
\tt{Vanishing Thm:} 
Giving unitary $(\kk,M)$-module $(\rho,L)$, finite dimensional $K_1$-module 
$(\tau,F)$,\\
a). if $\rho(C) \neq \tau(C)$, 
  $H^j(\kk,M; L\otimes F)=0$, for $\forall j$;\\
b). If $\rho(C) = \tau(C)$, 
  $H^j(\kk,M; L\otimes E) =\Hom_K(\wedge^j(\kk/\mm),L\otimes F)$.
\end{frame}

\begin{frame}[t]
(A1) $\kk$ hase a real form $\kk_1$ and $(K_1,M)$ is a symmetric pair.\\
(A2) $V = \bigoplus_j L_j$, $L_j$ irreducible unitarizable $(\kk,M)$-module.\\
(A3) $V$ has a $(\gg_0,M)$-invariant non-degenerate Hermition form $\inn{}{}$
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\begin{frame}[t]
\tt{More:} $L=L(E)$ is unitary highest weight $(\kk_1,M)$-module, $E$ with h.w. $\lambda$ 
$\lambda+\rho$ integral and regular,
$s(\lambda+\rho)$ dominant $\Rightarrow$ 
$\exists W\in \bigwedge^{l(s)}\kk/\mm$, 
$\Hom_M(W,L\otimes F^*) = \Hom_M(W,E\otimes F^*)$
\end{frame}

\begin{frame}[t]
\tt{Transfer of unitary structure:} Suppose $(\pi, V)$-irr. unitary. $(\gg',K')$
module. $\exists$ operator $T$, s.t. 
$\pi(\theta\theta'(X))=T \pi(X)T^{-1}$,
$T\pi(m)=\pi(m)T\quad \forall X\in\gg, m\in M$
and $L^{\oo_k^+} \subset V^T$.
$\Rightarrow \inn{v}{w} = \innc{Tv}{w}$ 
is the form making $\Gamma_W^jV$ unitary.  
\end{frame}

\begin{frame}[t]
\tt{Theorem:} When $V=L(F)$ be a unitarizable highest weight $(\gg',K')$-module.
$\theta(\oo^+) = \oo^+$, 
$\theta(k')f=k'f, \quad k'\in K', f\in F$
$\Gamma_W(V)$ is either zero or unitarizable.
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\begin{frame}[t]
\tt{Examples:} $(\gg,K)\simeq(\gg',K')\simeq Sp(2n,\bR)$.
$V=\theta^{0,m}(1)$ for dual pair $(O(0,m),Sp(2n,\bR))$, $m<n$.
Conjecture $\Gamma_{W_{p,q}}(V) \simeq \theta^{p,q}(1^{\xi,\eta})$
\end{frame}

\begin{frame}[t]
\begin{center}
\large Go further. 
\end{center}
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\begin{frame}[t]
\begin{center}
\large Thank you!
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\end{frame}

\end{document}
